Originally Posted by tangerineowl

Looking at your upper back I'd say you do need core work.

Can't quite tell because of your shirt, but it looks like your shoulders are slightly hunched upward? (arms/shoulders need to relax).

Core work would push your spine forward and your position at the bars lower and perhaps a little longer (stem swap).

Would find your body weight better balanced between fore-aft when riding, thus reduced shoulder pain.